Chapter 7.6, Problem 15E

Interpretation Introduction

Interpretation:

To show that frequency of small oscillations of amplitude $a\ll 1$ is $\text{ω}\approx 1-\frac{1}{16}{\text{a}}^2$

To show that formula for $\text{ω}$ is consistent with the exact result $\text{T = }2\text{π}\left(1+\frac{1}{16}{a}^2+O\left(a^4\right)\right)$

Concept Introduction:

The system equation for linear oscillator is $\ddot{\text{x}}\text{ + x = 0}$. If the system is perturbed by small perturbation constant, the system equation becomes $\ddot{\text{x}}\text{ + x + εh}\left(\text{x,}\dot{\text{x}}\right)\text{= 0}$

Here, $0<\epsilon \ll 1$, and $\text{h}\left(\text{x,}\dot{\text{x}}\right)$ is smooth function.

This system is known as weakly nonlinear oscillator.

The polar transformation of Cartesian co-ordinates is

$\begin{array}{l}\text{x = rcosθ}\\ \text{y = rsinθ}\end{array}$

The averaged equations for ${\text{r}}^{\prime }\text{ and r}{\varphi }^{\prime }$ are

${\text{r}}^{\prime }\text{=}\frac{\text{1}}{\text{2π}}{\displaystyle {\int }_{\text{0}}^{\text{2π}}\text{h}\left(\text{θ}\right)}\text{sinθdθ}\equiv \langle \text{h sinθ}\rangle$

$\text{r}{\varphi }^{\prime }\text{=}\frac{\text{1}}{\text{2π}}{\displaystyle {\int }_{\text{0}}^{\text{2π}}\text{h}\left(\text{θ}\right)}\text{cosθdθ}\equiv \langle \text{h cosθ}\rangle$

Where $\text{h}\left(\text{θ}\right)\text{=h}\left(\text{x,}\dot{\text{x}}\right)\text{=h}\left(\text{r cosθ},-\text{r sinθ}\right)$

The value of ${\text{r}}_{\text{0}}$ is calculated as:

${\text{r}}_0\text{=}\sqrt{{\left(\text{x}\left(\text{0}\right)\right)}^{\text{2}}\text{+}{\left(\dot{\text{x}}\left(\text{0}\right)\right)}^{\text{2}}}$

The value of ${\varphi }_0$ is calculated by:

${\varphi }_0={\tan }^{-}\left(\frac{\dot{\text{x}}\left(\text{0}\right)}{\text{x}\left(\text{0}\right)}\right)$

The exact formula for time period is $\text{T = }2\text{π}\left(1+\frac{1}{16}{a}^2+O\left(a^4\right)\right)$

The Taylor’s series expansion for ${\left(1-x\right)}^{-1}$ is

${\left(\text{1- x}\right)}^{\text{-1}}{\text{=1+x+x}}^{\text{2}}{\text{+x}}^{\text{4}}\text{+}\mathrm{......}$